Parts and Whole: We all know that parts are subsets of something, while whole refers to the complete thing. The idea of fractions is used in mathematics to represent parts and wholes. We understand that a fraction is a portion of a whole. This issue is concerned with defining and determining the connection between fractions and their whole. This post will go through how we can utilise models to split a large number of equal pieces.

This topic’s solved problems are provided to help students practise the concepts. Students will learn to develop, determine, and explain if a given fraction and an area representation correspond.

Definition of Parts and Whole

Whole means something complete. We can consider an entire pizza, a whole watermelon, a whole birthday cake, etc. The parts are the portions of a whole. For example, if we divide a certain quantity of pulses into four equal parts, each part so obtained is said to be one-fifth \(\left({\frac{1}{5}} \right)\) of the whole quantity of the pulses.

Similarly, if we divide a watermelon into four equal parts, each part is one-fourth \(\left({\frac{1}{4}} \right)\) of the whole watermelon. Now, if we eat three parts of these four equal parts, one part is left, and we say one-fourth \(\left({\frac{1}{4}} \right)\) of the watermelon is left.

Now, the eaten parts are \(\left({\frac{3}{4}} \right).\) The numbers \(\left({\frac{1}{4}} \right),\left({\frac{1}{5}} \right),\left({\frac{3}{4}} \right)\) considered above, each represents a part of the whole quantity, which are called fractions. A fraction is a quantity that expresses a part of the whole.

Examples of Parts and Whole

Let us make the concept more clear: Draw a circle with any radius. Divide it into three equal parts.

Now, observe the figure again.

Learn About Types of Fractions

If we shade two parts of the three equal parts, we say two-thirds \(\left({\frac{2}{3}} \right)\) of the circle is shaded, and one-third \(\left({\frac{1}{3}} \right)\) of the circle is not. Let us take another example of a number. It can give you a little bit more precise concept.
Now, have a look at the figure:

Number \(9\) is a whole number, and we divide it into two parts that are \(5\) and \(4.\) So, number \(9\) is a whole, and \(5\) and \(4\) are the parts of number \(9.\) In other words, we can say if we add two numbers, \(5\) and \(4.\) we will get the whole as \(9.\)

Fraction as a Part of a Whole

Let us know what are fractions exactly. The term fraction expresses a numerical quantity that is part of a whole thing. Suppose we have a large piece of rectangular-shaped paper, and we cut it into \(9\) equal parts and colour four parts in light blue and five parts in light grey.

Then each part of the paper is only \({\frac{1}{9}^{{\text{th}}}}\) of the total paper. We can write the part of blue coloured paper as \({\frac{4}{9}^{{\text{th}}}}\) part of paper and grey part of the paper as \({\frac{5}{9}^{{\text{th}}}}\) part of the paper. Here, \(\frac{1}{9},\frac{4}{9},\frac{5}{9}\) are all fractions.

Parts of a Fraction

Let us have a look at the image.

A fraction has three parts:

Numerator: The upper half of a fraction that expresses the number of portions you have. In the shown example, the numerator is \(5.\)

Denominator: The bottom half of a fraction expresses the number into which we divide the whole object. In the shown example, the denominator is \(9.\)

Bar Line: The line that divides the numerator and denominator. As shown in the image above, we read the whole fraction as \(‘5\) by \(8’.\) Likewise, we read \(\frac{1}{4}\) as \(‘1\) by \(4’,\) and \(\frac{3}{4}\) as \(‘3\) by \(4’.\)

Thus, in a fraction \(\frac{a}{b},a\) is the numerator of the fraction and, \(b\) is the denominator of the fraction.

\({\text{Fraction}} = \frac{{{\text{ Numerator }}}}{{{\text{ Denominator }}}}\)

For example, in \(\frac{1}{9},1\) is known as the numerator, and \(9\) is known as the denominator, for \(\frac{4}{9},4\) is known as the numerator, and \(9\) is known as the denominator. Similarly, for \(\frac{5}{9},5\) is known as the numerator, and \(9\) is known as the denominator. We don’t always deal with whole objects in our regular life. We deal with parts or portions of whole objects very often. To quantify them, we need fractions.

Types of Fractions

There are mainly 3 types of fractions:

Proper Fractions

A fraction is said to be a proper fraction if its value is less than one. In a proper fraction, the value of the numerator always has a lesser value compared to the value of the denominator.

The condition of the proper fraction is \({\text{numerator < denominator}}{\text{.}}\)
For example, \(\frac{5}{9},\frac{2}{7},\frac{{15}}{{19}},\frac{{500}}{{900}},\) etc.

Improper Fractions

A fraction is an improper fraction if its value is greater than or equal to one. In an improper fraction, the value of the numerator always has a larger value or equals as compared to the value of the denominator.

The condition of the improper fraction is the \({\text{numerator}} \geqslant {\text{denominator}}.\)

For example, \(\frac{{15}}{9},\frac{{27}}{7},\frac{{150}}{{19}},\frac{{500}}{9},\)etc.

Mixed Fractions

A mixed fraction is a fraction combination of whole and part. The value of a mixed fraction is always greater than one. It has a natural number along with a proper fraction, as shown below:

Solved Examples – Parts and Whole

Q.1. Express the following mixed fractions to improper fractions.
(a) \(2\frac{3}{4}\) (b) \(7\frac{1}{9}\)
Ans: Given mixed fractions are \(2\frac{3}{4}\) and \(7\frac{1}{9}.\)
To convert the mixed fractions to an improper fraction, multiply the denominator with the whole and add to the numerator keeping the same denominator.
Mixed fractions are converted to an improper fraction as follows:
\(\frac{{\left({{\text{whole}} \times {\text{denominator}}} \right){\text{ + numerator}})}}{{{\text{ denominator }}}}\)
\( \Rightarrow 2\frac{3}{4} = \frac{{\left({2 \times 4} \right) + 3}}{4} = \frac{{8 + 3}}{4} = \frac{{11}}{4}\)
\( \Rightarrow 7\frac{1}{9} = \frac{{\left({7 \times 9} \right) + 1}}{9} = \frac{{63 + 1}}{9} = \frac{{64}}{9}\)

Q.2. To make a cake, \(1\frac{1}{2}\) cups of sugar are needed. How many cups of sugar is necessary for baking \(6\) cakes?
Ans: Given, to make a cake, \(1\frac{1}{2}\) cups of sugar is needed.
We can find the number of cups of sugar required to bake \(6\) cakes by multiplying the number of cakes with the sugar necessary for one cake.
\( = 1\frac{1}{2} \times 6\)
We can convert the above-mixed fraction to an improper fraction by multiply the denominator with the whole and add to the numerator keeping the same denominator.
Mixed fractions are converted to an improper fraction as follows:
\(\frac{{\left({{\text{whole}} \times {\text{denominator}}} \right){\text{ + numerator}})}}{{{\text{ denominator }}}}\)
\( \Rightarrow 1\frac{1}{2} = \frac{{\left({1 \times 2} \right) + 1}}{2} = \frac{3}{2}\)
The amount of sugar required for making \(6\) cakes \( = \frac{3}{2} \times 6 = \frac{{18}}{2} = 9\)
Hence, to make the \(6\) cakes, we require \(9\) cups of sugar.

Q.3. Look at the square and express the blue portion in a fraction.

Ans: Let us first divide this square into equal divisions.

So, we can see that there are total \(4 \times 4 = 16\) equal small squares.
In that,
Number of rose coloured squares \( = 8\)
Number of blue coloured squares \( = 4\)
Number of pink coloured squares \( = 2\)
Number of orange coloured squares \( = 1\)
Number of white coloured squares \( = 1\)
Hence, the total number of equal parts is \(16,\) and blue coloured parts are \(4.\) So, we can represent the blue coloured parts as \(\frac{4}{{16}}.\)

Q.4. Identify the improper fractions from below.
\(\frac{5}{{28}},\frac{{13}}{5},\frac{9}{{16}},\frac{{19}}{8},\frac{8}{5},\frac{{16}}{3},7\frac{3}{4}\)
Ans: We know that fractions with the numerator equal to or greater than the denominator are called improper fractions.
Therefore, the improper fractions are \(\frac{{13}}{5},\frac{{19}}{8},\frac{8}{5},\frac{{16}}{3}.\)

Q.5. Identify the mixed fractions from below.
\(\frac{{19}}{5},\frac{7}{{16}},4\frac{3}{4},\frac{{17}}{8},\frac{3}{5},\frac{{17}}{3},\frac{6}{{28}}\)
Ans: We know that the sum of a whole number and a fraction is known as a mixed fraction.
So, the mixed fraction is \(4\frac{3}{4}.\)

Summary

In the above article, we learned about the definition of parts and whole, fraction, improper fraction, and mixed fraction—the topic parts and whole are similar to the concept of fractions. We have seen how to represent a part of a whole using models and different pictures. We have discussed the relation between the parts and whole and solved some examples related to the topic.

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Frequently Asked Questions (FAQs) – Parts and Whole

Q.1. What are fractions? Explain with an example.
Ans: A fraction is a quantity that used to expresses a part of the whole. When we divide a whole into parts, then each part can be expressed as a fraction.
Example: \(\frac{2}{3}.\)

Q.2. What is numerator and denominator?
Ans: A fraction is a quantity that used to express a part of the whole. The upper half of a fraction that defines the number of portions you have is known as the numerator, and the bottom part of the fraction that represents the whole is known as the denominator.
In \(\frac{a}{b},a\) is the numerator, and \(b\) is the denominator.

Q.3. What are the parts and whole concept?
Ans: Whole means something complete. We can consider a whole pizza, a whole watermelon, a whole birthday cake, etc. The parts are the portions of a whole. For example, if we divide a certain quantity of pulses into four equal parts, each part so obtained is said to be one-fifth \(\left({\frac{1}{5}} \right)\) of the whole quantity of the pulses.

Q.4. What is part and whole example?
Ans: Number \(9\) is a whole number, and if we divide it into two parts, that are \(5\) and \(4.\) So, number \(9\) is a whole, and \(5\) and \(4\) are the parts of number \(9.\) In other words, we can say if we add two numbers \(5\) and \(4\) we will get the whole as \(9.\)

Q.5. What is the concept of the part and examples?
Ans: Suppose we have a large piece of rectangular-shaped paper, and we cut it into \(9\) equal parts and colour four parts in blue and five parts in red. Then each part of the paper is only \({\frac{1}{9}^{{\text{th}}}}\) of the total paper. We can write the part of blue coloured paper as \({\frac{4}{9}^{{\text{th}}}}\) part of paper and grey part of the paper as \({\frac{5}{9}^{{\text{th}}}}\) part of the paper. Here, \(\frac{1}{9},\frac{4}{9},\frac{5}{9}\) are all fractions.

Related Concepts:

Fraction in Simplest Form	Division of Fractions
Multiplication of Fractions	Conversion of Fractions
Addition and Subtraction of Fractions	Equivalent Fractions
Fraction Calculator	Proper and Improper Fractions

We hope this detailed article on parts and whole helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. Happy learning!